PillCam-SPECTRA Bleeding-Region Detection (PWDR) L1-512

Medical ImagingCapsule-endoscopic bleeding / polyp / mucosal-anomaly classification via thresholded chromophore recoveryδ=3 · standardL_DAG = 6.9📋 Stub — not mineable

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Unclaimed Principle — open for contribution

This Principle is declared in the catalog but has no reference solver, no pinned dataset, and is not registered on-chain. There is no reward pool. Submitting a cert against this Principle today will record the cert for reproducibility but pay zero PWM.

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Forward model E

S t a g e 1 (ana l y t i c a l, f r o m L 1 - 511) : I_{b} a c k (r, λ) = I_{0} (λ) \cdot R_{d} i f f u se (m u_{a} (r, λ), m u_{s}^{'} (r, λ), g, r h o); r eco v er (c_{H} b O 2, c_{H} b, c_{w} a t er, c_{l} i p i d, c_{m} e l anin) (r) an d m u_{s}^{'} (r, λ) . S t a g e 2 (d e t er mini s t i c t h r es h o l d) : l ab e l (r) = a r g ma x_{c} l a ssscor e_{c} l a ss (c, m u_{s}^{'}) p er d i scr e t e_{r} e a d o u t . t h r es h o l d_{f} u n c t i o n . J o in t : c l a ss i f i c a t i o n (r) < - t h r es h o l d (L 1 - 51 1_{r} eco v er (I_{b} a c k)) .

PillCam-SPECTRA Bleeding-Region Detection (PWDR): the v2 physical_with_discrete_readout wrapper around L1-511 analytical core. The forward operator is the composition (threshold_function . L1-511_forward): multispectral capsule frames -> recovered chromophore concentrations (c_HbO2, c_Hb, c_water, c_lipid, c_melanin)(r) via L1-511 -> categorical labels {normal, bleeding, polyp, suspected_neoplasia, mucosal_inflammation}(r) via piecewise-constant threshold function on chromophore concentrations and reduced scattering coefficient mu_s'. The threshold is locally Lipschitz-continuous on the underlying physical state with global Lipschitz constant ~0.10 (per the smoothed-softmax formulation in discrete_readout.threshold_function). Recovery is posed as the joint problem: (i) recover continuous physical state via L1-511 (analytical S1-S4); (ii) apply deterministic threshold to produce categorical readout (S1-S4 inherited via Lipschitz continuity). The gate_substitutions field populates only conservation_substitute (softmax simplex closure on categorical readout); kappa_substitute and h_substitute are null because the physics core retains its analytical kappa and h. Difficulty tier delta = 3 inherited from L1-511. Mismatch parameters extend the L1-511 set: threshold_calibration_uncertainty, between-region_smoothing_artifact, motion-induced_label_flicker, capsule_orientation_mucosal_assignment.

L-DAG

L.optical_source -> L.diffuse_reflectance -> L.beer_lambert -> L.detector_response -> L.chromophore_unmixing -> L.threshold_classifier -> int.spatial -> int.spectral

L.optical_sourceL.diffuse_reflectanceL.beer_lambertL.detector_responseL.chromophore_unmixingL.threshold_classifierint.spatialint.spectral

Well-posedness W

Existence:: true
Uniqueness:: conditional
Stability:: conditional
κ:: 80

Existence and uniqueness inherited from L1-511 analytical core; threshold function is piecewise-constant and Lipschitz-continuous on the underlying chromophore-concentration state, so existence and uniqueness lift from continuous to discrete readout under the standard categorical-readout argument (PWM_GATE_CLASSES_AND_SCALING.md Case A). Stability inherits L1-511's kappa_eff ~ 15 plus a small additive contribution C_threshold * |threshold_calibration_uncertainty| from per-class threshold-value uncertainty. The threshold-continuity_proof in discrete_readout demonstrates that boundaries between class regions form measure-zero hyperplanes in chromophore space, so misclassification probability scales linearly with chromophore-recovery error away from the boundary.

Solvability C

Solver class:: linear-operator + convex optimisation [L1-511 chromophore unmixing] + categorical-readout [argmax / smoothed-softmax threshold] | end-to-end deep neural prior [PillCamNet, GINet] with explicit physics-informed analytical-core regularization
Convergence rate q:: 2
Complexity:: O(H * W * N_wavelengths * N_chromophores) for stage 1 (inherited from L1-511) + O(H * W * N_classes) for stage 2 threshold evaluation; total ~stage-1-dominated

Specs (0)

No L2 specs registered yet for this principle.